6.1.4: Reasoning about Equations and Tape Diagrams (Part 1)

Lesson

Let's see how tape diagrams can help us answer questions about unknown amounts in stories.

Exercise \(\PageIndex{1}\): Algebra Talk: Seeing Structure

Find a solution to each equation without writing anything down.

\(x+1=5\)

\(2(x+1)=10\)

\(3(x+1)=15\)

\(500=100(x+1)\)

Exercise \(\PageIndex{2}\): Situations and Diagrams

Draw a tape diagram to represent each situation. For some of the situations, you need to decide what to represent with a variable.

1. Diego has 7 packs of markers. Each pack has \(x\) markers in it. After Lin gives him 9 more markers, he has a total of 30 markers.
2. Elena is cutting a 30-foot piece of ribbon for a craft project. She cuts off 7 feet, and then cuts the remaining piece into 9 equal lengths of \(x\) feet each.
3. A construction manager weighs a bundle of 9 identical bricks and a 7-pound concrete block. The bundle weighs 30 pounds.
4. A skating rink charges a group rate of $9 plus a fee to rent each pair of skates. A family rents 7 pairs of skates and pays a total of $30.
5. Andre bakes 9 pans of brownies. He donates 7 pans to the school bake sale and keeps the rest to divide equally among his class of 30 students.

Exercise \(\PageIndex{3}\): Situations, Diagrams, and Equations

Each situation in the previous activity is represented by one of the equations.

• \(7x+9=30\)
• \(30=9x+7\)
• \(30x+7=9\)

1. Match each situation to an equation.
2. Find the solution to each equation. Use your diagrams to help you reason.
3. What does each solution tell you about its situation?

Are you ready for more?

While in New York City, is it a better deal for a group of friends to take a taxi or the subway to get from the Empire State Building to the Metropolitan Museum of Art? Explain your reasoning.

Summary

Many situations can be represented by equations. Writing an equation to represent a situation can help us express how quantities in the situation are related to each other, and can help us reason about unknown quantities whose value we want to know. Here are three situations:

1. An architect is drafting plans for a new supermarket. There will be a space 144 inches long for rows of nested shopping carts. The first cart is 34 inches long and each nested cart adds another 10 inches. The architect want to know how many shopping carts will fit in each row.
2. A bakery buys a large bag of sugar that has 34 cups. They use 10 cups to make some cookies. Then they use the rest of the bag to make 144 giant muffins. Their customers want to know how much sugar is in each muffin.
3. Kiran is trying to save $144 to buy a new guitar. He has $34 and is going to save $10 a week from money he earns mowing lawns. He wants to know how many weeks it will take him to have enough money to buy the guitar.

We see the same three numbers in the situations: 10, 34, and 144. How could we represent each situation with an equation?

In the first situation, there is one shopping cart with length 34 and then an unknown number of carts with length 10. Similarly, Kiran has 34 dollars saved and then will save 10 each week for an unknown number of weeks. Both situations have one part of 34 and then equal parts of size 10 that all add together to 144. Their equation is \(34+10x=144\).

Since it takes 11 groups of 10 to get from 34 to 144, the value of \(x\) in these two situations is \((144-34)\div 10\) or 11. There will be 11 shopping carts in each row, and it will take Kiran 11 weeks to raise the money for the guitar.

In the bakery situation, there is one part of 10 and then 144 equal parts of unknown size that all add together to 34. The equation is \(10+144x=34\). Since 24 is needed to get from 10 to 34, the value of \(x\) is \((34-10)\div 144\) or \(\frac{1}{6}\). There is \(\frac{1}{6}\) cup of sugar in each giant muffin.

Glossary Entries

Definition: Equivalent Expressions

Equivalent expressions are always equal to each other. If the expressions have variables, they are equal whenever the same value is used for the variable in each expression.

For example, \(3x+4x\) is equivalent to \(5x+2x\). No matter what value we use for \(x\), these expressions are always equal. When \(x\) is 3, both expressions equal 21. When \(x\) is 10, both expressions equal 70.

Practice

Exercise \(\PageIndex{4}\)

Draw a square with side length 7 cm.

1. Predict the perimeter and the length of the diagonal of the square.
2. Measure the perimeter and the length of the diagonal of the square.
3. Describe how close the predictions and measurements are.

(From Unit 3.1.1)

Exercise \(\PageIndex{5}\)

Find the products.

1. \((100)\cdot (-0.09)\)
2. \((-7)\cdot (-1.1)\)
3. \((-7.3)\cdot (5)\)
4. \((-0.2)\cdot (-0.3)\)

(From Unit 5.3.2)

Exercise \(\PageIndex{6}\)

Here are three stories:

• A family buys 6 tickets to a show. They also pay a $3 parking fee. They spend $27 to see the show.
• Diego has 27 ounces of juice. He pours equal amounts for each of his 3 friends and has 6 ounces left for himself.
• Jada works for 6 hours preparing for the art fair. She spends 3 hours on a sculpture and then paints 27 picture frames.

Here are three equations.

• \(3x+6=27\)
• \(6x+3=27\)
• \(27x+3=6\)

1. Decide which equation represents each story. What does \(x\) represent in each equation?
2. Find the solution to each equation. Explain or show your reasoning.
3. What does each solution tell you about its situation?

Exercise \(\PageIndex{7}\)

Here is a diagram and its corresponding equation. Find the solution to the equation and explain your reasoning.

Figure \(\PageIndex{1}\)

Exercise \(\PageIndex{8}\)

1. Plot these points on the coordinate plane:

\(A=(3,2),\: B=(7.5,2),\: C=(7.5, -2.5),\: D=(3,-2)\)

Figure \(\PageIndex{2}\)

2. What is the vertical difference between \(D\) and \(A\)?

3. Write an expression that represents the vertical distance between \(B\) and \(C\)?

(From Unit 5.2.6)